Greaterthanadditive

Greaterthanadditive, sometimes written as greater-than additive, is a term used to describe a property of functions or measures that aligns with the mathematical concept of superadditivity. In this sense, a function is greater-than additive if, for its domain closed under addition, it satisfies f(a + b) ≥ f(a) + f(b) for all admissible a and b. The term is informal; the standard mathematical name is superadditive. When the inequality is strict for some a and b, the function is said to be strictly greater-than additive.

Typical setting and definition: most commonly, f is defined on the nonnegative reals [0, ∞) or another additive semigroup, and the condition holds for all nonnegative a and b in the domain. If equality holds for all a and b, f is additive. Subadditive functions, by contrast, satisfy f(a + b) ≤ f(a) + f(b). The greater-than additive property implies that the function grows at least linearly under repeated addition.

Examples: simple instances include f(x) = c x with c ≥ 0, which is additive (hence GT-additive). Another example is f(x) = x^2 on [0, ∞), since (a + b)^2 ≥ a^2 + b^2 for all a, b ≥ 0. More generally, many convex, positively homogeneous functions exhibit a form of superadditivity on nonnegative domains.

Applications and relation: GT-additivity is used in economics to model increasing or convex returns to scale, in probability and analysis for growth and aggregation phenomena, and in optimization for understanding cumulative effects. It is related to, but distinct from, convexity and subadditivity, which describe different aggregation behaviors. See also superadditivity and subadditivity.